{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-quotient-ring",
  "semantic_family_id": "sf-finite-quotient-ring",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限環的商環"
  },
  "summary": {
    "zh-Hant": "有限環依proper雙邊理想的加法cosets形成商環，運算須與代表元選擇無關。"
  },
  "statement": {
    "zh-Hant": "a約b iff a-b屬I。classes為加法cosets；[a]+[b]=[a+b]、[a][b]=[ab]。完整核代表元獨立。本專案含一環要求1不等於0，因此排除I=R的退化zero ring。"
  },
  "formula": {
    "tex": "Q(R,I)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>Q</mi><mo>(</mo><mi>R</mi><mo>,</mo><mi>I</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "Q",
        "glyph": "Q"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        },
        {
          "type": "symbol",
          "name": "I",
          "glyph": "I"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3ba9a7db1f0613533192cf7645a50acefef6fd092c70fe75310a60df39d38f78"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限父環及已認證proper雙邊理想。"
    },
    {
      "id": "a2",
      "zh-Hant": "全部cosets正規化，運算逐代表元核well-defined。"
    }
  ],
  "symbols": [
    {
      "token": "Q",
      "role": "finite_quotient_ring",
      "role_zh": "有限商環",
      "scope": "representative_formula"
    },
    {
      "token": "I",
      "role": "two_sided_ideal",
      "role_zh": "proper雙邊理想",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-unital-ring",
      "title_zh": "有限含一環",
      "reason_zh": "父環完整雙表。"
    },
    {
      "id": "mko-finite-two-sided-ideal",
      "title_zh": "有限雙邊理想",
      "reason_zh": "同餘與良定性。"
    },
    {
      "id": "mko-equivalence-relation",
      "title_zh": "等價關係",
      "reason_zh": "商元素由理想同餘關係分組。"
    },
    {
      "id": "mko-quotient-set",
      "title_zh": "商集",
      "reason_zh": "載體為互異classes。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "以加法cosets形成載體。",
      "加乘運算逐代表元核同class。",
      "I=R另標記為不符合1不等於0慣例。"
    ],
    "common_misconception_zh": "挑代表元集合代替cosets，或未證良定就定義乘法。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理proper ideal；不宣稱quotient為域或整環。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入prime／maximal ideals。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-quotient-ring.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-10-31 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}